Proportionally increasing the width of a curved line
There are many ways to do this, but one that springs to mind is to construct your curve with a Curve modifier deformation, either of a long, thin plane subdivided along its length, or an array of smaller face-planes, merged in the Array modifier.

Then the curve's 'Radius' becomes available for adjustment. With Proprtional Editing, set to 'Connected Only', you can select one end or the other of the curve, with an appropriate falloff profile and size,  AltS scale the radius of the curve progressively from one end to the other.